An Imprimitivity Theorem for finite algebraic quantum groups

Let $\mathcal{G}$ be an algebraic quantum group and $\mathcal{U}$ a compact quantum subgroup. Given a left $\widehat{\mathcal{U}}$-module algebra A with unit, we can endow $A\otimes \mathcal{G}$ with a structure of a right $\widehat{\mathcal{U}}$-module algebra. The algebra of invariants for this action $(A\otimes \mathcal{G}{)}^{\widehat{\mathcal{U}}}$ has a left action of $\widehat{\mathcal{G}}$. We prove that for finite $\mathcal{G}$, $(A\otimes \mathcal{G}{)}^{\widehat{\mathcal{U}}}\mathrm{\#}\widehat{\mathcal{G}}$ is Morita equivalent to $A\mathrm{\#}\widehat{\mathcal{U}}$.